Bollobás–Riordan polynomial

The Bollobás–Riordan polynomial can mean a 3-variable invariant polynomial of graphs on orientable surfaces, or a more general 4-variable invariant of ribbon graphs, generalizing the Tutte polynomial.

History
These polynomials were discovered by.

Formal definition
The 3-variable Bollobás–Riordan polynomial of a graph $$G$$ is given by


 * $$R_G(x,y,z) =\sum_F x^{r(G)-r(F)}y^{n(F)}z^{k(F)-bc(F)+n(F)}$$,

where the sum runs over all the spanning subgraphs $$F$$ and
 * $$v(G)$$ is the number of vertices of $$G$$;
 * $$e(G)$$ is the number of its edges of $$G$$;
 * $$k(G)$$ is the number of components of $$G$$;
 * $$r(G)$$ is the rank of $$G$$, such that $$r(G) = v(G)- k(G)$$;
 * $$n(G)$$ is the nullity of $$G$$, such that $$n(G) = e(G)-r(G)$$;
 * $$bc(G)$$ is the number of connected components of the boundary of $$G$$.